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Inferring in absence of background
The solution of Eq.(4) depends, at least
in principle, on the assumption on the prior .
Taking a flat prior between 0 and 1, that models our
indifference on the possible values of before
we take into account the result of the experiment in which
successes were observed in trials, we get
(see e.g. [2]):

(12) 
some examples of which are shown in Fig. 1.
Figure:
Probability density function of the binomial parameter
, having observed successes in trials.[2]

Expected value, mode (the value of for which
has the maximum) and variance of this distribution are:
Eq. (13)
is known as
``recursive Laplace formula'',
or ``Laplace's rule of succession''.
Not that there is no magic if the formula
gives a sensible result even
for the extreme cases and for all values of
(even if !).
It is just a consequence of the prior: in absence of new information,
we get out what we put in!
From Fig. 1 we can see that for large numbers
(and with far from 0 and from )
tends to a Gaussian. This is just the reflex of the limit
to Gaussian of the binomial. In this large numbers limit
and
.
Subsections
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Giulio D'Agostini
20041213